The quantum-mechanical Coulomb propagator in an L2 function representation

The quantum-mechanical Coulomb propagator is represented in a square-integrable basis of Sturmian functions. Herein, the Stieltjes integral containing the Coulomb spectral function as a weight is evaluated. The Coulomb propagator generally consists of two parts. The sum of the discrete part of the spectrum is extrapolated numerically, while three integration procedures are applied to the continuum part of the oscillating integral: the Gauss–Pollaczek quadrature, the Gauss–Legendre quadrature along the real axis, and a transformation into a contour integral in the complex plane with the subsequent Gauss–Legendre quadrature. Using the contour integral, the Coulomb propagator can be calculated very accurately from an L$$^2$$ 2 basis. Using the three-term recursion relation of the Pollaczek polynomials, an effective algorithm is herein presented to reduce the number of integrations. Numerical results are presented and discussed for all procedures.

On Certain Properties and Applications of the Perturbed Meixner–Pollaczek Weight

This paper deals with monic orthogonal polynomials orthogonal with a perturbation of classical Meixner–Pollaczek measure. These polynomials, called Perturbed Meixner–Pollaczek polynomials, are described by their weight function emanating from an exponential deformation of the classical Meixner–Pollaczek measure. In this contribution, we investigate certain properties such as moments of finite order, some new recursive relations, concise formulations, differential-recurrence relations, integral representation and some properties of the zeros (quasi-orthogonality, monotonicity and convexity of the extreme zeros) of the corresponding perturbed polynomials. Some auxiliary results for Meixner–Pollaczek polynomials are revisited. Some applications such as Fisher’s information, Toda-type relations associated with these polynomials, Gauss–Meixner–Pollaczek quadrature as well as their role in quantum oscillators are also reproduced.

Permanent-Magnet Synchronous Motor Drive System Using Backstepping Control with Three Adaptive Rules and Revised Recurring Sieved Pollaczek Polynomials Neural Network with Reformed Grey Wolf Optimization and Recouped Controller

Owing to some nonlinear characteristics in the permanent-magnet synchronous motor (SM), such as nonlinear friction, cogging torque, wind stray torque, external load torque, and unmodeled systems, fine control performances cannot be accomplished by utilizing the general linear controllers. Thereby, the backstepping approach adopting three adaptive rules and a swapping function is brought forward for controlling the rotor motion in the permanent-magnet SM drive system to reduce nonlinear uncertainties effects. To improve the chattering phenomenon, the backstepping control with three adaptive rules using a revised recurring sieved Pollaczek polynomials neural network (RRSPPNN) with reformed grey wolf optimization (RGWO) and a recouped controller is proposed to estimate the internal collection and external collection torque uncertainties, and to recoup the smallest fabricated error of the appraised rule. In the light of the Lyapunov stability, the on-line parametric training method of the RRSPPNN can be derived through an adaptive rule. Furthermore, to obtain a beneficial learning rate and improve the convergence of the weights, the RGWO algorithm adopting two exponential-functional adjustable factors is applied to adjust the two learning rates of the weights. Then, the efficiency of the used controller is validated by test results.

A Rectified Reiterative Sieved-Pollaczek Polynomials Neural Network Backstepping Control with Improved Fish School Search for Motor Drive System

As the six-phase squirrel cage copper rotor induction motor has some nonlinear characteristics, such as nonlinear friction, nonsymmetric torque, wind stray torque, external load torque, and time-varying uncertainties, better control performances cannot be achieved by utilizing general linear controllers. The snug backstepping control with sliding switching function for controlling the motion of a six-phase squirrel cage copper rotor induction motor drive system is proposed to reduce nonlinear uncertainty effects. However, the previously proposed control results in high chattering on nonlinear system effects and overtorque on matched uncertainties. So as to reduce the immense chattering situation, we then put forward the rectified reiterative sieved-Pollaczek polynomials neural network backstepping control with an improved fish school search method to estimate the external bundled torque uncertainties and to recoup the smallest reorganized error of the evaluated rule. In the light of Lyapunov stability, the online parametric training method of the rectified reiterative sieved-Pollaczek polynomials neural network can be derived by utilizing an adaptive rule. Moreover, to improve convergence and obtain beneficial learning manifestation, the improved fish school search algorithm is made use of to readjust two fickle learning rates of the weights in the rectified reiterative sieved-Pollaczek polynomials neural network. Lastly, the effectuality of the proposed control system is validated by examination results.